gb_trees

General Balanced Trees

An efficient implementation of Prof. Arne Andersson's General
Balanced Trees. These have no storage overhead compared to
unbalanced binary trees, and their performance is in general
better than AVL trees.

This module considers two keys as different if and only if
they do not compare equal (==).

Data structure

There is no attempt to balance trees after deletions. Since
deletions do not increase the height of a tree, this should be OK.

Original balance condition h(T) <= ceil(c * log(|T|))
has been changed to the similar (but not quite equivalent)
condition 2 ^ h(T) <= |T| ^ c. This should also be OK.

Performance is comparable to the AVL trees in the Erlang book
(and faster in general due to less overhead); the difference is
that deletion works for these trees, but not for the book's
trees. Behaviour is logarithmic (as it should be).

DATA TYPES

gb_tree() = a GB tree

Functions

balance(Tree1) -> Tree2

Tree1 = Tree2 = gb_tree()

Rebalances Tree1. Note that this is rarely necessary,
but may be motivated when a large number of nodes have been
deleted from the tree without further insertions. Rebalancing
could then be forced in order to minimise lookup times, since
deletion only does not rebalance the tree.

delete(Key, Tree1) -> Tree2

Key = term()

Tree1 = Tree2 = gb_tree()

Removes the node with key Key from Tree1;
returns new tree. Assumes that the key is present in the tree,
crashes otherwise.

delete_any(Key, Tree1) -> Tree2

Key = term()

Tree1 = Tree2 = gb_tree()

Removes the node with key Key from Tree1 if
the key is present in the tree, otherwise does nothing;
returns new tree.

empty() -> Tree

Tree = gb_tree()

Returns a new empty tree

enter(Key, Val, Tree1) -> Tree2

Key = Val = term()

Tree1 = Tree2 = gb_tree()

Inserts Key with value Val into Tree1 if
the key is not present in the tree, otherwise updates
Key to value Val in Tree1. Returns the
new tree.

from_orddict(List) -> Tree

List = [{Key, Val}]

Key = Val = term()

Tree = gb_tree()

Turns an ordered list List of key-value tuples into a
tree. The list must not contain duplicate keys.

get(Key, Tree) -> Val

Key = Val = term()

Tree = gb_tree()

Retrieves the value stored with Key in Tree.
Assumes that the key is present in the tree, crashes
otherwise.

lookup(Key, Tree) -> {value, Val} | none

Key = Val = term()

Tree = gb_tree()

Looks up Key in Tree; returns
{value, Val}, or none if Key is not
present.

insert(Key, Val, Tree1) -> Tree2

Key = Val = term()

Tree1 = Tree2 = gb_tree()

Inserts Key with value Val into Tree1;
returns the new tree. Assumes that the key is not present in
the tree, crashes otherwise.

is_defined(Key, Tree) -> bool()

Tree = gb_tree()

Returns true if Key is present in Tree,
otherwise false.

is_empty(Tree) -> bool()

Tree = gb_tree()

Returns true if Tree is an empty tree, and
false otherwise.

iterator(Tree) -> Iter

Tree = gb_tree()

Iter = term()

Returns an iterator that can be used for traversing the
entries of Tree; see next/1. The implementation
of this is very efficient; traversing the whole tree using
next/1 is only slightly slower than getting the list
of all elements using to_list/1 and traversing that.
The main advantage of the iterator approach is that it does
not require the complete list of all elements to be built in
memory at one time.

keys(Tree) -> [Key]

Tree = gb_tree()

Key = term()

Returns the keys in Tree as an ordered list.

largest(Tree) -> {Key, Val}

Tree = gb_tree()

Key = Val = term()

Returns {Key, Val}, where Key is the largest
key in Tree, and Val is the value associated
with this key. Assumes that the tree is nonempty.

map(Function, Tree1) -> Tree2

Function = fun(K, V1) -> V2

Tree1 = Tree2 = gb_tree()

maps the function F(K, V1) -> V2 to all key-value pairs
of the tree Tree1 and returns a new tree Tree2 with the same set of keys
as Tree1 and the new set of values V2.

next(Iter1) -> {Key, Val, Iter2} | none

Iter1 = Iter2 = Key = Val = term()

Returns {Key, Val, Iter2} where Key is the
smallest key referred to by the iterator Iter1, and
Iter2 is the new iterator to be used for
traversing the remaining nodes, or the atom none if no
nodes remain.

size(Tree) -> int()

Tree = gb_tree()

Returns the number of nodes in Tree.

smallest(Tree) -> {Key, Val}

Tree = gb_tree()

Key = Val = term()

Returns {Key, Val}, where Key is the smallest
key in Tree, and Val is the value associated
with this key. Assumes that the tree is nonempty.

take_largest(Tree1) -> {Key, Val, Tree2}

Tree1 = Tree2 = gb_tree()

Key = Val = term()

Returns {Key, Val, Tree2}, where Key is the
largest key in Tree1, Val is the value
associated with this key, and Tree2 is this tree with
the corresponding node deleted. Assumes that the tree is
nonempty.

take_smallest(Tree1) -> {Key, Val, Tree2}

Tree1 = Tree2 = gb_tree()

Key = Val = term()

Returns {Key, Val, Tree2}, where Key is the
smallest key in Tree1, Val is the value
associated with this key, and Tree2 is this tree with
the corresponding node deleted. Assumes that the tree is
nonempty.

to_list(Tree) -> [{Key, Val}]

Tree = gb_tree()

Key = Val = term()

Converts a tree into an ordered list of key-value tuples.

update(Key, Val, Tree1) -> Tree2

Key = Val = term()

Tree1 = Tree2 = gb_tree()

Updates Key to value Val in Tree1;
returns the new tree. Assumes that the key is present in the
tree.

values(Tree) -> [Val]

Tree = gb_tree()

Val = term()

Returns the values in Tree as an ordered list, sorted
by their corresponding keys. Duplicates are not removed.